Relevant background publications include the following:
McDowell, A. and Fukushima, E., “Ultracompact NMR: 1H Spectroscopy in a Subkilogram Magnet,” Applied Magnetic Resonance 35 (1), 185-195, 2008. This reference demonstrates NMR spectroscopy in a compact permanent magnet with nanoliter-volume samples.
Blümich, Bernhard, et al., “Mobile NMR for Geophysical Analysis and Materials Testing,” Petroleum Science 6 (1), 1-7, 2009. This reference shows a compact NMR spectrometer that employs a Halbach magnet design.
Chmurny, Gwendolyn N. and Hoult, David I., “The Ancient and Honourable Art of Shimming,” Concepts in Magnetic Resonance Part A 2 (3), 131-149, 2005. This reference details the use of spherical harmonic function expansions in shimming.
Raich, H. and Blümler, P., “Design and Construction of a Dipolar Halbach Array with a Homogeneous Field from Identical Bar Magnets: NMR Mandhalas,” Concepts in Magnetic Resonance B: Magnetic Resonance Engineering 23B (1), 16-25, 2004. This reference details the use of Halbach-style magnets made from cubic magnets in nuclear magnetic resonance spectrometers.
Moresi, Giorgio and Magin, Richard, “Miniature Permanent Magnet for Table-top NMR,” Concepts in Magnetic Resonance Part B: Magnetic Resonance Engineering 19B (1), 35-43, 2003. This reference discloses efforts to render the field inside Halbach arrays more homogeneous for NMR applications using flat pole pieces. It also mentions a ridged design configuration.
Danieli, Ernesto, “Mobile Sensor for High Resolution NMR Spectroscopy and Imaging,” Journal of Magnetic Resonance 198, 80-87, 2009. This reference discloses efforts to render the field more homogeneous using magnets placed within the primary Halbach array.
Keim, Thomas A., “Intentionally Non-orthogonal Correction Coils for High-homogeneity Magnets,” U.S. Pat. No. 4,581,580, 1986. Discloses the use of a set of shim coils capable of producing multiple spherical harmonics through variation of the specified set of applied currents. A given coil within the set can contribute to more than a single spherical harmonic function.
Golay, M. J. E., “Homogenizing Coils for NMR Apparatus,” U.S. Pat. No. 3,622,869, 1971. Discloses the use of homogenizing coils for optimization of magnetic fields that consist of electrical conductors affixed to electrically insulating plates and placed parallel and adjacent to magnetic pole pieces.
Kabler, Donald J., Gang, Robert E., and Reeser, Jr., William O., “Magnetic Field Shim Coil Structure Utilizing Laminated Printed Circuit Sheets,” U.S. Pat. No. 3,735,306, 1973. Discloses field homogenizing coils constructed with printed circuit sheets placed parallel and adjacent to pole pieces in a separate module.
U.S. Pat. No. 4,682,111, 1987 to Hughes discloses the use of shaped pole pieces for improving the homogeneity of the static magnetic field.
Rose N. E., “Magnetic Field Correction in the Cyclotron”, Phys. Rev. 53, 715-719, 1938. Describes ridged pole pieces for use in homogenizing magnetic fields in cyclotrons.
O'Donnell, Matthew, et al., “Method for Homogenizing a Static Magnetic Field Over an Arbitrary Volume,” U.S. Pat. No. 4,680,551, (issued on July 14) 1987. Discloses selection of shimming currents based on magnetic field mapping and a weighted least-squares calculation.
In a nuclear magnetic resonance (NMR) experiment, a sample is placed under the influence of a biasing static magnetic field, which partially aligns the sample's nuclear-spin magnetic moments. The moments precess in the static field at a frequency, called the Larmor frequency, which is proportional to the field strength. The magnetic moments of the sample can be manipulated by applying a transverse radio frequency (RF) magnetic field at the Larmor frequency. By observing the reaction of the sample to the RF field, insight into the chemical composition of the sample can be gained. The power of NMR as an analytical method may be largely a function of how well the characteristics of the applied magnetic fields can be controlled.
The practice of shimming magnetic fields (rendering the fields more uniform) has existed since the earliest days of NMR and originally used thin pieces of metal physically placed behind source magnets to adjust the positions of those magnets in order to refine the magnetic field. More modern shimming techniques use electro-magnetic coils. Conventional magnetic resonance spectrometers commonly use shimming coils disposed on substantially cylindrical coil forms. The use of shim coils in compact NMR devices has proved difficult primarily due to space restrictions that may not accommodate traditional shim coil systems, which can have many layers. The space available inside a main magnet in many such devices may be too small to accommodate a typical set of shimming coils whose individual elements are each designed predominantly to address one and only one geometrical aspect or geometrical component of the residual inhomogeneity of the main magnetic field.
FIGS. 1A, 1B, and 1C compare the main biasing field and sample-tube configurations of typical high-field spectrometer designs with a design for compact magnet systems that is based on the cylindrical Halbach array. The arrows labelled B indicate the main magnetic field direction. No shimming measures are shown in the figures. FIG. 1A schematically shows the superconducting field coils of the high-field magnet, an inserted cylindrical sample tube, and the field, B, produced by the coils. The magnetic field within the sample volume is aligned along the common symmetry axis of the coils and the tube.
FIGS. 1B and 1C show the same sample tube inserted into a cylindrical Halbach magnet array, which produces a field, B, perpendicular to the symmetry axis of the tube. This particular Halbach array is composed of eight magnets in a circular arrangement placed around the tube, with the magnetization vectors of the magnets (shown as arrows) perpendicular to the tube's symmetry axis. The field inside the Halbach array is quite uniform for some applications, but can be too inhomogeneous for some high-resolution NMR experiments.
In order to substantially reduce the inhomogeneity of a magnetic field, it may be helpful to have independent control over different geometrical aspects of the field inhomogeneity. In many magnetic resonance applications, the main magnetic field is strongly polarized along a specified direction, which we take to be the z-axis in a Cartesian reference frame whose origin is at some fixed point. The Larmor frequency of magnetic spins located at a point in space is determined by the magnitude of the field at that point, which in reasonably homogeneous fields is very well approximated by the z-component of the field, Bz. One can expand Bz as a scaled sum of functions,Bz(x,y,z)=B0+Σkckfk(x,y,z),where k is a variable (or a number of variables) used to index the various functions, ƒk, in the set, and where x, y, and z are Cartesian or other spatial coordinates defining positions within a volume enclosing at least part of the sample. B0 is the large and spatially uniform part of the field, and the coefficients, ck, quantify different components of the field inhomogeneity. Such sets of functions, for example x, z, xy, ½(z2−y2) are said to be orthogonal (with respect to a specified scalar product of functions) if the scalar product between two functions that are not the same is zero. A common scalar product between two functions is the integral,k1|k2≡∫VW(x,y,z)f*k1(x,y,z)fk2(x,y,z)dV, where V denotes a volume relevant to the functions over which the integral is calculated, where the star denotes complex conjugation, and where W denotes a weighting function defined on the volume, which quantifies how important the volume element at (x, y, z) is in its contribution to the integral.
For example, commonly, an expansion in spherical harmonic functions is used, where the functions arefn,m(x,y,z)=Nn,mPn,m(cos θ)exp(imφ),where θ=tan−1(√{square root over (x2+y2)}/z) and φ=tan−1(y/x), where Pn,m denotes a Legendre polynomial or associated Legendre function, and where Nn,m are normalization factors. In this case W(x, y, z)=δ(√{square root over (x2+y2+z2)}−1), where δ denotes the Dirac delta function, and the functions are said to be “orthogonal over the unit sphere.” Sometimes, real-valued linear combinations of the complex-valued spherical harmonic functions are used instead.
If, in addition, the scalar product between each function ƒk and itself is equal to 1, then the set of functions is said to be orthonormal.